Gaming Guru

Why Probability Doesn't Tell You What'll Probably Happen Next

Every gambler should have a haruspex. Then we wouldn't need probabilities to
tell us what'll probably happen next. Which is just as well, because that's
not what probabilities do.

Probabilities are good for predicting long-run averages. In the casino, for
instance, the house's edge on a row of slot machines -- found by combining possible
payouts and their probabilities of occurrence -- will forecast its monthly profitability
dependably enough to take to the bank. This, because a typical installation
gets tens of millions of pulls, all with roughly comparable bets, during any
meaningful accounting period. It says next to nothing about how Max and Mabel
will fare in today's three-hour session.

Most players spend their casino careers in the "short run." They
get nowhere near the amount of action necessary for results to average out.
And for extreme longshots such as jackpots, numbers of trials undergone are
inevitably far less than what's confusingly often referred to as a "cycle"
or "statistically correct cycle" so they shouldn't "expect"
to hit at all.

Of course, individual solid citizens are well advised to tailor their play
according to the laws of probability. For instance, by following Expert Strategy
at video poker or Basic Strategy at blackjack. But doing so won't help anticipate
where they'll be at the end of any particular session. In fact, they'd really
like to finish someplace where probability alone says they won't.

To get an idea why probability has more to do with the uncertain than the likely,
picture a three-spin parlay on a single number at roulette. The chance of hitting
any number on a spin in a double-zero game is one out of 38. That of hitting
any number in three successive spins is one out of 38 multiplied by itself three
times. In case you don't have your abacus or slide rule handy, that's one out
of 54,872.

Would 54,872 starts give you confidence in hitting the parlay? Is it realistic
to figure on good luck bringing home the bacon earlier? Should you be prepared
to persevere a lot longer to reap your reward? Lacking a personal haruspex,
you might turn to computer simulation to help ponder puzzles like these.

I ran two such simulations for this purpose. In each, virtual players bet on
the three-spin parlay until it hit, then started over and repeated the process
until they completed 1,000 winning series. Outcomes showed numbers of tries
to achieve success. The separate simulations yielded somewhat different results,
as would be expected because 1,000 repeats are ample to suggest trends but still
exhibit a moderate degree of volatility.

In the first simulation, the average number of tries required to hit the parlay
was 55,255, with the smallest being 61 and the largest 501,299. The second battery
showed an average of 56,470 tries, ranging from 1 to 533,777. The distribution
of results is summarized in the accompanying table.

tries before hitting parlay

run 1

run 2

up to 100

3

2

101 to 500

86

78

501 to 1,000

5

6

1,001 to 5,000

67

53

5,001 to 10,000

67

76

10,001 to 25,000

175

150

25,001 to 54,872

234

242

54,873 to 75,000

90

129

75,001 to 100,000

96

92

100,001 to 150,000

101

97

over 150,000

76

75

In both simulations, the greatest concentrations of hits were from 25,000 to
75,000 tries: 324 of the 1,000 sequences in the first run, 371 in the second.
However, 94 and 86 blissful bettors won within only 1,000 attempts, while 76
and 75 sad souls needed over 150,000 rounds. None of the series took the theoretically
"expected" 54,872 bids. The closest on either side were 54,843 and
54,917 in one case and 54,759 and 54,950 in the other.

So, what does "one out of 54,872" really tell the person who steps
up to take a few shots? Not much! Here's how that veritable versemonger, Sumner
A Ingmark, characterized the conundrum:

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